# Confidence interval for the restricted mean in survival analysis

What is the recommended way of calculating confidence intervals for the restricted mean in survival analysis? Here are some example data, taken from the paper

Chris Barker (2009), The Mean, Median, and Confidence Intervals of the Kaplan–Meier Survival Estimate—Computations and Applications, The American Statistician, Volume 63, Issue 1, 2009.

Note that, despite the title, the paper does not list any confidence intervals for the mean. R (3.0.2) can calculate an estimate of the restricted mean, but only calculates a confidence interval for the median, not for the mean:

> library(survival)
> time = c(10, 15, 23, 30, 35, 52, 100)
> dead = c(1, 1, 1, 1, 1, 1, 0)
> fit = survfit(Surv(time, dead) ~ 1)
> print(fit, print.rmean=TRUE)
Call: survfit(formula = Surv(time, dead) ~ 1)

records      n.max    n.start     events     *rmean *se(rmean)     median    0.95LCL    0.95UCL
7.0        7.0        7.0        6.0       37.9       10.7       30.0       15.0         NA
* restricted mean with upper limit =  100

(Here the restricted mean is estimated to be 37.9, based on a cutoff of 100.) Stata (13.0), on the other hand, happily calculates a confidence interval:

|    no. of  restricted
|  subjects        mean      Std. Err.    [95% Conf. Interval]
-------------+-------------------------------------------------------------
total |         7    37.85714(*)   10.73888      16.8093     58.905

(*) largest observed analysis time is censored, mean is underestimated

This seems to be based on a normal approximation, i.e., estimated mean ± 1.96 standard errors. In R, one can calculate this more or less manually, using numbers extracted from the fit:

fit.table = summary(fit, rmean=TRUE)\$table # or, preferably, rmean=100
rmean = fit.table[["*rmean"]]                               # Estimate restricted mean
rmean.se = fit.table[["*se(rmean)"]]                        # Estimated standard error
rmean + c(-1,1) * qnorm(.05/2, lower.tail=FALSE) * rmean.se # 95% CI
[1] 16.80932 58.90497

Is this a good way of calculating confidence intervals for restricted means (at least if the cutoff for the restricted mean is fixed in advance)? Or are there better alternatives?

• Vignette for survRM2 package: illustrates how to use the functions in the package to compare two groups with respect to the restricted mean survival time. – Fiorella Mar 29 '17 at 22:39